Numerical solution of minimax problems of optimal control,part 1

This paper contains general transformation techniques useful to convert minimax problems of optimal control into the Mayer-Bolza problem of the calculus of variations [Problem (P)]. We consider two types of minimax problems: minimax problems of Type (Q), in which the minimax function depends on the state and does not depend on the control; and minimax problems of Type (R), in which the minimax function depends on both the state and the control. Both Problem (Q) and Problem (R) can be reduced to Problem (P).For Problem (Q), we exploit the analogy with a bounded-state problem in combination with a transformation of the Jacobson type. This requires the proper augmentation of the state vectorx(t), the control vectoru(t), and the parameter vector , as well as the proper augmentation of the constraining relations. As a result of the transformation, the unknown minimax value of the performance index becomes a component of the parameter vector being optimized.For Problem (R), we exploit the analogy with a bounded-control problem in combination with a transformation of the Valentine type. This requires the proper augmentation of the control vectoru(t) and the parameter vector , as well as the proper augmentation of the constraining relations. As a result of the transformation, the unknown minimax value of the performance index becomes a component of the parameter vector being optimized.In a subsequent paper (Part 2), the transformation techniques presented here are employed in conjunction with the sequential gradient-restoration algorithm for solving optimal control problems on a digital computer; both the single-subarc approach and the multiple-subarc approach are discussed.This research was supported by the National Science Foundation, Grant No. ENG-79-18667, and by Wright-Patterson Air Force Base, Contract No. F33615-80-C3000. This paper is a condensation of the investigations reported in Refs. 1–7. The authors are indebted to E. M. Coker and E. M. Sims for analytical and computational assistance.     

Dynamic modification of the capillary wall for electrophoretic separations of small ions

Separations of small ions were carried out under nonequilibrated conditions using capillaries treated with NaOH, HCl, or tris(hydroxymethyl)aminomethane (Tris) prior to analysis. For separations of benzoic acid isomers or acids and amines under weakly acidic conditions, capillaries flushed with 0.1 M NaOH and subsequently with running buffers prior to analysis were used. Separations of six benzoic acid isomers were accomplished in 4 min in 1 mM phosphate buffers, pH 4.01, containing 2.5 mM hydroxypropyl-beta-cyclodextrin. Without additives, the separation of biological amines and acids were also achieved in 10 min at pH 4.01. Capillaries treated with 0.1 M HCl prior to analysis were tested in separations of six phenols in 5 mM Tris solutions at pH 7.0. As a result of small electrophoretic mobilities of phenols against a small electroosmotic flow, resolution was optimized. We also found that reproducibility was improved using capillaries treated with HCl. The relative standard deviations of migration mobility of phenols were less than 1%, which were smaller than those obtained using capillaries treated with 0.1 M NaOH or Tris.     

Optimizing separation conditions for polycyclic aromatic hydrocarbons in micellar electrokinetic chromatography

We report the separation of polycyclic aromatic hydrocarbons (PAHs) using 0.1% poly(ethylene oxide) (PEO) in micellar electrokinetic chromatography (MEKC). In the presence of PEO, adsorption of PAHs on the capillary wall was reduced, leading to better resolution and reproducibility. Effects of tetrapentylammonium iodide (TPAI), dextran sulfate (DS), methanol, and sodium lauryl sulfate (SDS) on the separation of PAHs were elucidated. In terms of resolution and speed, DS, compared to TPAI, is a better additive for separation of PAHs. When using 0.1% PEO solution containing 45% methanol, 50 mM SDS, and 0.02% DS, separation of 10 PAHs containing 2 to 5 benzene rings was accomplished in less than 12 min at 15 kV in a commercial CE system. The method has also been tested for separating seven PAHs with high quantum yields when excited at 325 nm using a He-Cd laser. Unfortunately, separation of the seven PAHs was not achieved and sensitivity diminished under the same conditions. To optimize sensitivity, resolution and speed, a stepwise technique in MEKC has been proposed. The seven PAHs were resolved in 35 min at 15 kV when separation was performed in 0.1% PEO solution containing 35 mM SDS, 40% methanol and 0.02% DS for 2 min, and subsequently in 0.1% PEO solution containing 20 mM SDS, 50% methanol, and 0.02% DS.     

Insertion Reactions of Carbon Disulfide into Mg‐C and Mg‐N Bonds: Syntheses and Structural Determinations of Mg(S2CY)2(THF)n(Y=NEt2, NiPr2, iPr,Ph) and BrMg(S2CZ)(THF)3(Z=Ph,iPr)

The insertion reaction of CS₂ with Mg(NR₂)₂ (R= Et, ⁱPr), MgR′₂ (R′= Et, Ph) and R″MgBr (R″= ⁱPr, Ph) respectively lead solid products, Mg(S₂CNR₂)₂(THF)_n ( 1 : R= Et, n=2; 2 : R= ⁱPr, n=1), Mg(S₂C′R)₂(THF)₂ ( 3 : ′R= Et, 4 : ′R= Ph), BrMg(S₂C″R) (THF)₃ ( 5 : ″R= ⁱPr, 6 : ″R= Ph) in which the inserted carbon disulfides act as terminal chelating ligands. These compounds were characterized with ¹H, ¹³C NMR, IR spectroscopy, mass spectrometry, elemental analyses, and X‐ray crystallography.     

A 0.7-2-GHz self-calibrated multiphase delay-locked loop

A 0.7-2-GHz precise multiphase delay-locked loop (DLL) using a digital calibration circuit is presented. Incorporating with the proposed digital calibration circuit, the mismatch-induced timing error among multiphase clocks in the proposed DLL can be self-calibrated. When the calibration procedure is finished, the digital calibration circuit can be turned off automatically to save power dissipations and reduce noise generations. A start controlled circuit is proposed to enlarge the operating frequency range of the DLL. Both the start-controlled circuit and the calibration circuit require an external reset signal to ensure the correctness of the calibration after temperature,operating frequency, and power supply voltage are settled. This DLL with the digital calibration circuit has been fabricated in a 0.18-/spl mu/m CMOS process. The measured results show the DLL exhibits a lock range of 0.7-2 GHz while the peak-to-peak jitter and rms jitter is 18.9ps and 2.5 ps at 2 GHz, respectively. When the calibration procedure is completed and the DLL operates at 1 GHz, the maximum mismatch-induced timing error among multiphase clocks is reduced from 20.4 ps (7.34 degree) to 3.5 ps (1.26 degree).     

An automatic coefficient design methodology for high-order bandpass sigma-delta modulator with single-stage structure

An automatic coefficient design method for synthesis of bandpass sigma-delta modulators (BPSDMs) is presented in this brief. Single-stage BPSDM structures, cascade-of-resonator with distributed feedback, cascade-of-resonator with distributed feedforward, and a new structure with lower coefficient spread, are all used to fit the synthesized coefficients. The automatic coefficient design method is realized in an easy-to-use computer program. Even for inexperienced designers, reliable and high-tolerance BPSDM coefficients for various applications can be automatically and efficiently generated. The methodology covers many design concerns including BPSDM coefficient tolerance for circuit component mismatch, design tradeoffs among in-band noise suppression, oversampling ratio, modulator order and quantizer bit number. Finally, design examples with orders of 6 and 8, and quantizer bit number of 1-bit and 3-bit, respectively, are used for the verification of the proposed automatic coefficient design method.     

Triple-frequency planar monopole antenna for side-feed communication device on GSM/DCS/PCS operation

A novel side-feed planar monopole antenna capable of triple-frequency operation at about 900, 1800 and 1900 MHz is presented. The planar monopole antenna occupies a small area of 6/spl times/31 mm, and is easily made using thin copper. The planar monopole antenna can be side-fed and mounted perpendicularly to the main circuit board of a communication device so that it offers a novel design with a free degree of feed point so as to save device space, resulting in a low profile to the system ground plane. In addition, the obtained impedance bandwidths of the proposed antenna at about 900, 1800 and 1900 MHz can cover the GSM (890-960 MHz), DCS (1710-1880 MHz) and PCS (1850-1990 MHz) bands.